2696 CHAPTER 79. INCLUDING STOCHASTIC INTEGRALS

Theorem 79.1.7 Assume the above conditions, 79.1.1 - , 79.1.7 along with the progressivemeasurability condition 79.1.2. Also assume there is at most one solution to 79.1.8 where

q(t, ·)≡∫ t

0ΦdW

Then there exists a P measurable u such that also z is progressively measurable

Bu(t,ω)−Bu0 (ω)+∫ t

0z(s,ω)ds =

∫ t

0f (s,ω)ds+B

∫ t

0ΦdW

where for each ω, z(·,ω) ∈ A(u(·,ω) ,ω). The function Bu(t,ω) = B(u(t,ω)) for a.e. t.Here

Φ ∈ Lα

(Ω;L∞

([0,T ] ,L2

(Q1/2U,W

)))∩L2

([0,T ]×Ω,L2

(Q1/2U,W

)),α > 2

and u0 ∈ L2 (Ω,W ), f ∈ Lp′ ([0,T ]×Ω;V ′).

The following corollary comes right away from the above and uniqueness for fixed ω .

Corollary 79.1.8 In the situation of Theorem 79.1.7, change the conditions on Φ. Insteadof letting

Φ ∈ Lα

(Ω;L∞

([0,T ] ,L2

(Q1/2U,W

)))∩L2

([0,T ]×Ω,L2

(Q1/2U,W

))assume that Φ ∈ L2

([0,T ]×Ω;L2

(Q1/2U,W

))and that t → Φ(t,ω) is continuous into

L2(Q1/2U,W

). Then there exists a unique solution to the integral equation of Theorem

79.1.7.

Proof: Let τm = inf{

t : ∥Φ(t,ω)∥L2(Q1/2U,W) > m}. Then Φτm is uniformly bounded

above by m and limm→∞ τm = ∞. Hence Φτm is in the necessary space for the conclusionof Theorem 79.1.7 to hold. Letting m→ ∞ and using uniqueness, one finds a solution tothe integral equation.

79.2 Replacing Φ With σ (u)One can replace Φ with σ (u) provided B maps W one to one onto W ′. This includes themost common case of a Gelfand triple in which B = I and V ⊆ H = H ′ ⊆ V ′. We alsoneed to assume that A is defined pointwise as described below rather than possibly havingmemory terms involved.

Theorem 79.2.1 In the situation of Theorem 79.1.7, suppose 1 - 79.4.2 and progressivemeasurability condition 79.1.2 but A defined pointwise,

A(u,ω)(t) = A(u(t,ω) , t,ω)

and suppose f is progressively measurable and is in Lp′(

Ω;Lp′ ([0,T ] ;V ′))

. Assume

⟨Bu,u⟩ ≥ δ ∥u∥2W